# Simplified Model of Nonlinear Landau Damping

## Abstract

The nonlinear interaction of a plasma wave with resonant electrons results in a plateau in the electron distribution function close to the phase velocity of the plasma wave. As a result, Landau damping of the plasma wave vanishes and the resonant frequency of the plasma wave downshifts. However, this simple picture is invalid when the external driving force changes the plasma wave fast enough so that the plateau cannot be fully developed. A new model to describe amplification of the plasma wave including the saturation of Landau damping and the nonlinear frequency shift is proposed. The proposed model takes into account the change of the plasma wave amplitude and describes saturation of the Landau damping rate in terms of a single fluid equation, which simplifies the description of the inherently kinetic nature of Landau damping. A proposed fluid model, incorporating these simplifications, is verified numerically using a kinetic Vlasov code.

- Authors:

- Publication Date:

- Research Org.:
- Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 959129

- Report Number(s):
- PPPL-4426

TRN: US1005725

- DOE Contract Number:
- DE-AC02-09CH11466

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; AMPLIFICATION; AMPLITUDES; DISTRIBUTION FUNCTIONS; ELECTRONS; KINETICS; LANDAU DAMPING; PHASE VELOCITY; PLASMA WAVES; SATURATION; Nonlinear Theories, Quaslinear Theories, Viasov Equetion

### Citation Formats

```
N. A. Yampolsky and N. J. Fisch.
```*Simplified Model of Nonlinear Landau Damping*. United States: N. p., 2009.
Web. doi:10.2172/959129.

```
N. A. Yampolsky and N. J. Fisch.
```*Simplified Model of Nonlinear Landau Damping*. United States. doi:10.2172/959129.

```
N. A. Yampolsky and N. J. Fisch. Thu .
"Simplified Model of Nonlinear Landau Damping". United States.
doi:10.2172/959129. https://www.osti.gov/servlets/purl/959129.
```

```
@article{osti_959129,
```

title = {Simplified Model of Nonlinear Landau Damping},

author = {N. A. Yampolsky and N. J. Fisch},

abstractNote = {The nonlinear interaction of a plasma wave with resonant electrons results in a plateau in the electron distribution function close to the phase velocity of the plasma wave. As a result, Landau damping of the plasma wave vanishes and the resonant frequency of the plasma wave downshifts. However, this simple picture is invalid when the external driving force changes the plasma wave fast enough so that the plateau cannot be fully developed. A new model to describe amplification of the plasma wave including the saturation of Landau damping and the nonlinear frequency shift is proposed. The proposed model takes into account the change of the plasma wave amplitude and describes saturation of the Landau damping rate in terms of a single fluid equation, which simplifies the description of the inherently kinetic nature of Landau damping. A proposed fluid model, incorporating these simplifications, is verified numerically using a kinetic Vlasov code.},

doi = {10.2172/959129},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Thu Jul 16 00:00:00 EDT 2009},

month = {Thu Jul 16 00:00:00 EDT 2009}

}